which leads to his optimal measurement \(\Pi ^*_1\)  as in (REF ).
The attacker, after observing the true hypothesis (\(H_1\)  or \(H_0\) ) on the target systems, obtains the quantum state \(|\varphi \rangle \in \mathcal {H}\) . Based on his observation on the true density state (\(\rho _1\)  or \(\rho _0\) ), the attacker designs quantum noisy operations \({E}^1,E^0\in B(\mathcal {H})\)  acting upon the received state \(|\varphi \rangle \)  to create a distorted quantum state \(|\varphi ^{\prime }\rangle \) . According to [1]}, we formulate the resulting noisy density operators \(E^0, E^1\)  as the `operator-sum' representation as follows:
\(\begin{aligned}\mathcal {E} &= \lbrace (E^0, E^1)\in {B}(\mathcal {H})^{\otimes 2}: \sum _{k}{E^l_kE^{l\dagger }_k} = \mathbf {1},\;\;l\in \lbrace 0,1\rbrace \rbrace .\end{aligned}\) 
